a framework based on compressed manifold modes for robust

Eurographics Workshop on 3D Object Retrieval (2017)I. Pratikakis, F. Dupont, and M. Ovsjanikov (Editors)

A framework based on Compressed Manifold Modes for robust localspectral analysis

S. Haas1, A. Baskurt1, F. Dupont2 and F. Denis2

1Univ Lyon, INSA de Lyon, LIRIS, F-69621, LYON, France2Univ Lyon, Univ Lyon 1, LIRIS, F-69622, LYON, France

AbstractCompressed Manifold Modes (CMM) were recently introduced as a solution to one of the drawbacks of spectral analysis ontriangular meshes. The eigenfunctions of the Laplace-Beltrami operator on a mesh depend on the whole shape which makesthem sensitive to local aspects. CMM are solutions of an extended problem that have a compact rather than global supportand are thus suitable for a wider range of applications. In order to use CMM in real applications, an extensive test hasbeen performed to better understand the limits of their computation (convergence and speed) according to the compactnessparameter, the mesh resolution and the number of requested modes. The contribution of this paper is to propose a robust choiceof parameters, the automated computation of an adequate number of modes (or eigenfunctions), stability with mutltiresolutionand isometric meshes, and an example application with high potential for shape indexation.

Categories and Subject Descriptors (according to ACM CCS): I.3.3 [Computer Graphics]: Picture/Image Generation—Line andcurve generation

1. Introduction

Spectral analysis of meshes relies on the study of the eigenvec-tors of specifically defined mesh operators. It is used to solve awide variety of problems such as mesh compression, segmentation,smoothing, watermarking or correspondence. The Laplacian oper-ator is the well-known operator that is defined on a function f as

∆ f = div(−−→grad( f )) and specifically in R2 as ∆ f = ∂

2 f∂x2 + ∂

2 f∂y2 . The

Laplace-Beltrami operator (LBO) is its extension on a manifolds,which, in our case, is a smooth surface inside R3. The eigenfunc-tions of the Laplacian operator on R2 (i.e. the solutions of equation∆ f = λ f where λ is a real number) form the basis for the spectralanalysis of functions that are defined on R2. The eigenfunctions ofthe LBO have the same role for functions that are defined on thesurface.

The LBO discrete counterpart is thus used to perform spectralanalysis on a 3D triangle mesh [Tau95]. The eigenfunctions of thediscrete LBO are usually referred to as the Manifold Harmonic Ba-sis (MHB) of the considered mesh [Lev06], [VL08] and [RBG∗09].So, the surface decomposition over the MHB can be seen as theFourier spectrum of the mesh and as such allow its spectral analy-sis. It is independent of the extrinsic shape properties and invariantunder natural shape deformation [Rus07]. They have already beenstudied for a wide range of applications such as building shape de-scriptors [RWP06], point descriptors and shape signature [SOG09],shape matching [OMMG10] or shape segmentation [SOCG10].

The Compressed Manifold Modes (CMM) were introduced byNeumann et al. in order to overcome some limitations of globalspectral analysis [NVT∗14]. This paper presents a frameworkbased on the CMM tool and demonstrates how local spectral anal-ysis can be used in many applications.

Our contributions are: a set of tool parameters and mesh prop-erties that demonstrate robustness and are thus suitable for appli-cations; a simple algorithm to automatically compute an adequatenumber of CMM functions for a given shape; and we demonstrateour algorithm stability for varying resolutions of the same shapeand isometric deformations.

As example application, we propose a coarse matrix comparisonthat shows very promising results in shape matching.

2. Background, previous and related work

MHB eigenfunctions present a major drawback: they are global anddepend on the whole shape. It is thus difficult to interpret them.Moreover, they are sensitive to local modifications of the mesh suchas holes and noise that appear in scanned meshes. That global as-pect also hampers partial shape recognition. In order to overcomethese limitations, T. Neumann et al. introduced the CompressedManifold Basis (CMB) and its components are called the Com-pressed Manifold Modes (CMM). The MHB functions are the so-lutions ϕk of the equation

∆ϕk =−λkϕk,k ∈ N,λk ∈ R

c© 2017 The Author(s)Eurographics Proceedings c© 2017 The Eurographics Association.

DOI: 10.2312/3dor.20171054

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S. Haas, A. Baskurt, F. Dupont & F. Denis / A framework based on CMM for robust local spectral analysis

with ∆ representing the LBO. Usually, a set of K eigenfunctionscorresponding to the smallest eigenvalues λk is considered. TheCMM are the solutions of the generalized eigenvalues problemthat induces sparsity by introducing the `1-norm and a parameterµ ∈ R+:

minϕk

K

∑k=1〈ϕk|∆ϕk〉+

µN‖ϕk‖1 such that 〈ϕk|ϕ j〉= δk j

where δk j is the Kronecker delta that enforces orthogonality of theCMM or eigenfunctions, and N is the number of vertices of themesh. These functions have a compact support whose relative sizeis controlled with parameter µ. When µ = 0, we have the originalproblem the solution of which is the MHB; larger µ values yieldmore compact solutions.

3. CMM computation speed and convergence

3.1. Available implementations and test

The CMM computation algorithm proposed by Neumann et al. isbased on reformulating the original problem and solving it with thealternating direction method of multipliers (ADMM) as describedin [BPC∗11]. They claim that, even though the minimization prob-lem is not convex due to the orthogonality constraints, local minimaare reached and sufficient for practical applications. Python codehas been made available to reproduce some of their results. Anaccelerated version of the algorithm was later designed by Hous-ton [Hou15]. He claims this new algorithm is about 47% fasterthan the original one on average, and provides Matlab code to testit. Finally, a new paradigm regarding the `1-norm discretization haslately been promoted by Bronstein et al. [BCKS16]. The minimiza-tion problem is modified into a sequence of eigendecompositionones, which avoids non-convex optimization and thus achieves adrastic speedup in runtime. We tested the original algorithm and itsaccelerated counterpart in a python environment. Our goal was toidentify the tool limitations and check its applicability.

3.2. Impact of parameters on convergence

We evaluated both the original algorithm and its accelerated versionusing various shapes. For some typical cases, we also changed themesh resolution. Each mesh was then tested using different com-binations of the compactness parameter µ and the number K of re-quested eigenfunctions. The conditions of our test case are givenbelow:

• 22 meshes ranging from 453 to 74764 vertices.• 10 values for the compactness parameter ranging from 1 to 1000.• The number of requested eigenfunctions ranged from 5 to 40.• Every mesh has been tested with every possible combination of

K\µ parameters.• Computation was performed on a PC with an Intel i7-4710HQ

CPU @ 2.5GHz with 8Go RAM.

Table 1 shows the number of converging cases for the original al-gorithm. Results for the accelerated algorithm are very similar. Theconvergence criterion is fully described in [NVT∗14]. Non conver-gence happens when the 15000 iterations limit is reached. The algo-rithm converged in less than 50% of the test cases and it clearly ap-pears that non convergence happens when the number of requested

K\µ 1 2 5 10 20 50 100 200 500 10005 11 15 20 22 21 22 22 21 19 16

10 0 0 4 11 12 20 21 19 18 1620 0 0 0 0 0 9 11 16 14 1640 0 0 0 0 0 0 0 5 10 13

Total number of converging meshes among 22 for the K\µcombination.

Table 1: Convergence with the original CMM algorithm

eigenfunctions is large and the support is not compact enough. Thisis very probably due to the orthogonality constraint that is easierto fulfill while the eigenfunctions can have approximately sepa-rate supports. Speed was acceptable (less than 1 minute to get theCMM) in most converging cases when the number of requestedeigenfunctions was 20 or less.

4. Finding an adequate number of eigenfunctions

An important objective of our work was to design an algorithm thatautomatically determines the number of eigenfunctions that bestfit the application needs. Indeed, automation is essential for a toolin practical applications: it is not acceptable to rely on the userto fine tune various parameters in order to achieve its goal everytime the tool is called. The choices that we present below are basedon an extensive set of experiments on usual indexation databases:SHREC15 [LZC∗15], SCAPE [ASK∗05], FAUST [BRLB14], theBunny [TL94], Armadillo and Bimba models. The algorithm stop-ping criterion was designed with a shape matching application inmind.

4.1. Method

The same compactness parameter is used for all meshes (µ = 20).It is the control key to the number of eigenfunctions. The higherit is, the more eigenfunctions one gets for a given shape. Its finetuning should be done once according to the targeted type of appli-cation. It must be remembered that higher compactness parametervalues yield more eigenfunctions and might increase drastically thecomputation time.

Let us then define vertex coverage: a vertex from the mesh issaid covered by the CMM when at least one eigenfunction exceedsa preset percentage T hV of its absolute maximum value on thatvertex. Next, we consider the mesh is covered when at least a givenpercentage T hM of the vertices are covered.

The algorithm starts computing 1 eigenfunction and increasesthis number until the mesh is covered. Every step is initializedwith the previously found eigenfunctions and one additional con-stant function; the resulting eigenfunctions are then tested againstthe stopping criterion. The process starts over until the criterionis reached. Both thresholds can be adjusted according to the typeof desired application. For the application that we present at theend of this paper, the values that gave overall the best results wereT hV = 0.1 and T hM = 0.95.


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Figure 1: Algorithm progression

Figure 2: Effect of compactness parameter changes

4.2. Results

On average, for all our test cases, we get about 8 eigenfunc-tions with the selected compactness parameter. On the SHREC15dataset, the minimum is 3 eigenfunctions and the maximum is 19.More than 90% of the meshes yield between 4 and 12 eigenfunc-tions. Two examples in the Figure 1 show the algorithm progres-sion. A vertex is colored when it is covered. When different eigen-functions overlap, the color of the vertex corresponds to the onewith the highest value. These examples present an interesting ex-traction of the intrinsic global shape aspect. Changing either T hVor T hM threshold only modifies the number of steps of the algo-rithm that keeps adding new eigenfunctions until stopped. On theother hand, for a given mesh, compactness parameter changes havean impact on the eigenfunctions themselves as can be seen in theFigure 2. The compactness parameter can be adapted to the de-sired application type. However, it must be kept coherent with themesh resolution (the greater compactness, the finer details are cap-tured which requires higher resolution) and fit the application exactneeds.

In all the tests we made so far with our choice of parameters, theeigenfunction supports have always proved to be connex when themeshes do not have too many holes.

5. Stability over two mesh transformations

The process stability has been tested when a mesh is subsampledwith the idea of speeding up the process for a given shape by chang-ing its resolution. In order to address moving shapes, stability wasalso tested on isometrically deformed shapes.

5.1. Effects of subsampling

We tested subsampling on the Armadillo and the Bimba modelsusing Alliez and Desbrun algorithm [AD01]. For the Bimba, westarted with the original mesh that comprises 75000 vertices andsubsampled it successively to 30000, 9000, 4000 and 1200 vertices.

Figure 3: Stability with multiresolution shapes

For the Armadillo, the 61400 vertices original model was subsam-pled to 25500, 8400, 3400 and 900 vertices. As can be seen on theFigure 3, subsampling keeps the shape general aspect while remov-ing details; these modifications did not change the output of the al-gorithm that shows robustness with respect to the mesh density andlevel of details. For large meshes, this very interesting result effec-tively promotes the idea of subsampling to overcome the compu-tation time limitation. When the compactness parameter is µ = 20,1000 vertices is a good target provided that this level of subsam-pling is coherent with the shape itself. With a non-optimized pythonalgorithm on a PC with an Intel i7-4710HQ CPU @ 2.5GHz with8Go RAM, computation on the original Armadillo took 245.7 sec.whereas it was only 1.44 s for the 900 vertices subsampled mesh.From unacceptable computation delays in many applications, wedrop to perfectly reasonable speed.

5.2. Isometric deformation

Figure 4: Stability with isometric deformation

We evaluated our method on mesh databases that comprise dif-ferent poses of the same shape. This was done with the SHREC15and SCAPE datasets. For example, the SCAPE dataset contains 70different positions of the same human subject, and our algorithmfound very similar solutions: 66% showed one configuration (withthe same number of eigenfunctions identically located on the shapeand found in the same order), and two other close configurationswere found for 17% and 10% of the remaining meshes (they in-clude one additional eigenfunction). The other results were minorvariants of the 3 main configurations (e.g. the eigenfunctions arelocated in a different order from the original configuration). Someexamples are given in the Figure 4. These very promising resultsshow the stability of the method regarding isometric deformations.


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6. Application

Figure 5: Two sample groups from the SHREC15 database

We present here a simple method based on the eigenfunctioncoverage found by our previous algorithm, and propose coarseshape matching results based on it.

Our method first normalizes the eigenfunctions that have beenpreviously found (their values are thus comprised between -1 and1, their maximum being always 1). Then, we associate every ver-tex of the mesh with the function whose value is highest at it. Theregions that we obtain have always proved to be connected andwe order them according to their eigenfunction discovery rank. Wethen build a symmetrical boolean matrix that describes the regioncontiguity.

A coarse matrix comparison was run on the SHREC15 dataset:we grouped together meshes with the same matrix. Two clustersof meshes that were obtained with this comparison are shown inFigure 5. Although these results do not compare yet with state-of-the-art algorithms, the intrinsic simplicity of the tool makes it anencouraging new research direction for shape matching and index-ation.

7. Conclusion and future work

The Compressed Manifold Modes that have been recently intro-duced by Neumann et al. seem an interesting tool that addressesone of the major drawbacks of spectral analysis in shape match-ing, segmentation and other typical applications. We performed anadvanced study of that tool and found a robust set of parametersthat demonstrate algorithmic convergence at an acceptable speed.From these results, we proposed an algorithm that automaticallycomputes an adequate number of CMM eigenfunctions that prop-erly cover a given shape. We also showed that this method is stableand yields the same eigenfunctions regardless of the resolution andisometric deformation of the mesh, and gave a simple applicationwith promising outcomes. There are several directions for our fu-ture work on the subject:

• try the very recent proposition from Bronstein et al. and evaluatethe performance gain for our method;

• apply our method to movements and study its applicability toanalyze moving shapes.

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a framework based on compressed manifold modes for robust

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